Let me try to address some of your questions. My answers may not be as detailed as possible, since my experience is with Charter/Spectrum, but some of it carries over.
That depends. If you look at the TV Everywhere Availability support article, it will show you which networks you may receive with your Xfinity credentials. If the networks you want are not listed, then you will need a CableCARD tuner.
Now for the part I'm not certain about: Channels is incompatible with DRM flagged channels. Some providers protect more channels than others. I know Verizon/Frontier FiOS flag almost no channels, and some providers like Cox flag almost everything. In my area it was a toss up, but slowly more and more channels were being flagged.
Their 6 tuner model has been shelved. Their older 3 tuner model (HDHR3-CC) is still available, as is a refurbished (HDHR3-CC-R) model. (If you go to their store via their website, you need to select HDHomeRun Hardware, then scroll to the bottom. You will see an entry for the Prime, but clicking it will make it disappear. However, if you click on the ADD TO CART button, it will put it in a cart for ordering. Yes, their site is broken.)
Hopefully someone will have better answers for you than I provided above. I have roughly heard that Xfinity is pretty decent, meaning they have fewer DRM flagged channels than most.
In general, there is no limit. For a hardware tuner, you are limited by the device. The Prime can record/view 3 concurrent channels. If you are using your credentials and using internet-based TV Everywhere streams, then the limit is your internet connection and drive I/O bandwidth. However, if you are using the TV Everywhere streams, keep in mind that they count towards your data cap, if you have one.
While a Raspberry Pi-based system can work well, and for many users it does, I personally feel that a proper desktop might be the best option. I have found an Intel NUC or Gigabyte Brix to be more than sufficient. You can usually get one a few generations old that will serve you for many years.
I specifically switched to Channels from my previous DVR software solely for the excellent remote implementation. My upstream bandwidth is 10Mbps, and I had no problems with 720p/4Mbps streams. I personally found that combination to give enough fidelity, as well as not tax a busy wifi network where I was. (Hospital wifi is only slightly better than free hotel wifi.)